
where \(\sigma _{\mu \nu }=\nabla _{\langle {\mu }}u_{\nu \rangle }\) , \(D=u^{\mu }\partial _{\mu }\) , and \(\nabla ^{\mu }=\Delta ^{\mu \nu }\partial _{\nu }\)  are symmetric trace-less shear tensor, temporal and spatial counterparts of the total
space-time derivative respectively. Next, we use Eq. (REF ) in Eq. (REF ) and (REF ) to construct \(\phi \)  in order to calculate the first-order field correction coefficients defined below.
Usually in the next step one replaces all temporal derivatives in Eq. (REF ) by spatial derivatives using hydrodynamic equations of motion.
However, following the works of [1]}, [2]}, [3]} we do not carry out the above exercise
as is usually done for the conventional N-S theory; instead, we keep the time derivatives in the first-order corrections. Also, as per [1]}, [2]}, [3]} the first-order thermodynamic field corrections in a general frame and with arbitrary interactions can be written as:
\(\delta n^{(1)} =& \nu _{1}\frac{DT}{T} + \nu _{2} \left(\partial \cdot u\right) +\nu _{3} D\tilde{\mu }~,\\\delta \epsilon ^{(1)} =& \varepsilon _{1}\frac{DT}{T} + \varepsilon _{2} \left(\partial \cdot u\right) +\varepsilon _{3} D\tilde{\mu }~,\\\delta P^{(1)} =& \pi _{1}\frac{DT}{T} + \pi _{2} \left(\partial \cdot u\right) +\pi _{3} D\tilde{\mu }~,\\W^{(1)\mu } =& \theta _{1}\left(\frac{\nabla ^{\mu }T}{T} - Du^{\mu }\right) +\theta _{3} \nabla ^{\mu }\tilde{\mu }~,\\V^{(1)\mu } =& \gamma _{1}\left(\frac{\nabla ^{\mu }T}{T} - Du^{\mu }\right) +\gamma _{3} \nabla ^{\mu }\tilde{\mu }~.\) 
